Geophys. J. Int. (1998) 133, 726–740
Modelling anisotropic wave propagation in oceanic inhomogeneous
structures using the parallel multidomain pseudo-spectral method
Shu-Huei Hung and Donald W. Forsyth
Department of Geological Sciences, Brown University, Providence, 02912-1846 RI, USA. E-mail: [email protected]
Accepted 1997 December 9. Received 1997 December 9; in original form 1997 May 22
SU MM A RY
We have developed a multidomain pseudo-spectral method for 3-D anisotropic
seismic waveform modelling to run in a high-performance parallel computer. The
numerical scheme solves acoustic and elastic wave equations formulated by velocity
and stress for fluid layers and solid spaces, respectively. Spatial variations of velocity
and stress fields propagating in general anisotropic and heterogeneous media are
calculated by high-accuracy Fourier and Chebyshev differential operators. The
fourth-order Runge–Kutta method is employed for time-marching of the wavefields.
The domain decomposition technique divides the full space into several subdomains
bounded by discontinuous features with steep velocity gradients. The propagating
wavefronts are separately calculated in each subdomain, then wavefield matching
procedures subject to the continuity condition for the velocity and traction are imposed
at the interface of subdomains. Non-periodic boundary conditions at the free surface
and the non-reflected edge at the bottom of the model space are adequately represented
by expansion in Chebyshev polynomials. Parallelizing the serial algorithm effectively
reduces the computation time. Data locally distributed in many processors for the
parallel program provides the large memory storage needed for a 3-D model. The
waveforms calculated by this numerical scheme show accuracy comparable with a
known analytical solution. The numerical experiments indicate the effectiveness of the
multidomain approach in modelling seismic energy partitioning into reflected, refracted
and transmitted waves at velocity discontinuities such as the crust/water interface
and in generating synthetic seismograms at seafloor affected by water reverberations.
We simulate seismic wave propagation in isotropic and anisotropic elastic media with
hexagonal, orthorhombic and monoclinic symmetries, which represent possible symmetry systems in the mantle and crust. The propagating wavefields possess observable
differences in wavefront geometry, phase arrivals, disturbed amplitude and shear-wave
splitting. The azimuthal dependence in traveltimes and variability in synthetic waveforms demonstrates that fine details of anisotropic velocity structures can be resolved
by comparing simulated full waveforms with observed seismograms recorded at various
distances and azimuths.
Key words: anisotropic wave propagation, Chebyshev–Fourier method, full waveform
modelling, multidomain, parallel computation.
1
I NTR O D UC TIO N
Seismological evidence indicates that the interior of the Earth
is inhomogeneous and anisotropic on scales ranging from a
few to thousands of kilometres (e.g. Nishimura & Forsyth
1989; Crampin & Lovell 1991; Montagner & Tanimoto 1991).
Heterogeneity is mainly caused by seismic wave velocity varying with the composition and physical properties of the Earth.
Anisotropy is usually associated with the alignment of voids
726
or fluid-filled cracks (e.g. Crampin & Booth 1985; Blackman
& Kendall 1997), and/or the lattice preferred orientations of
minerals such as olivine and pyroxene in response to shear
deformation (e.g. Nicolas & Christensen 1987; Blackman et al.
1996). Seismic waves radiated from their source can generate
transmitted, reflected, converted and diffracted waves when
incident on continuous planar or curved or discontinuous
structures with velocity gradients. They can also be split when
they enter anisotropic media and then travel with different
© 1998 RAS
Parallel multidomain pseudo-spectral method
speeds. The polarization and velocity of the split waves depend
on the propagation direction, the compressional or shearing
type of wave motion and the anisotropic properties of the
media.
Because seismic wave propagation in both heterogeneous
and anisotropic structures is complicated, synthetic seismograms are needed to understand systematically the variability
of full waveforms propagating in a set of hypothetical velocity
models. This is essential for the further development of the
inverse problem in the determination of robust geological
structures. Many approaches have been developed to calculate
synthetic seismograms and anisotropic seismic wave propagation,
such as asymptotic ray theory (e.g. Chapman & Drummond
1982; Kendall & Thomson 1993) and the reflectivity method
(e.g. Booth & Crampin 1983). However, based on the highfrequency assumption, ray-tracing methods are valid only
for smooth structures with weak anisotropy and fail to
calculate accurately inhomogeneous waves in media with
large velocity jumps such as the water/crust discontinuity. The
reflectivity method works only for laterally homogeneous,
layered anisotropic structures.
Discrete solution methods such as finite differences (e.g.
Boore 1972), finite elements (e.g. Lysmer & Drake 1971) and
pseudo-spectral methods (e.g. Fornberg 1975) are in principle
suitable for more general anisotropic and inhomogeneous
structures with 3-D variations, and are capable of solving the
most general wave-motion-governing equation with no need
for simplification. They track the wavefields in time, and hence
any change of wavefronts in a complicated anisotropic and
heterogeneous medium can be clearly visualized. Moreover,
broad-band full waveforms that include inhomogeneous head
or surface waves and the effects of ray collapse and amplitude
singularity at caustics can be adequately simulated if a proper
numerical method is used. These advantages of numerical
modelling provide a great aid in the interpretation of the
observed arrival phases. However, the potentially intensive
computational work has hindered the broad application of
numerical solutions in modelling 3-D waveforms in large-scale
problems such as teleseismic studies. With currently available
computational power, these solutions are becoming increasingly
realistic.
The goal of this paper is to present an accurate and efficient
numerical method to simulate seismic wave propagation
in 3-D, generalized inhomogeneous and anisotropic media,
particularly in oceanic structures that have been studied using
ocean-bottom seismometers (OBSs). The technique we use is
the multidomain pseudo-spectral method. In this method, the
field variables are approximated by the global expansion of
a set of basis functions such as Fourier series. The spatial
derivatives at each grid point are calculated from all nodal
values along the differentiation direction and the expansion
coefficients of trigonometric interpolation rather than only
using a few neighbouring nodes and constant coefficients
in finite difference schemes. This global approximation of
derivatives minimizes numerical dispersion in the discrete
solutions and does not produce artificial anisotropy, which
is favourable for our interest in seismic anisotropy. Because
most methods for synthetic seismogram calculation utilize a
single-domain approach with global elastic properties and
a uniform grid discretization, they often have difficulties in
modelling structures with large velocity contrasts or changes
of physical properties across discontinuities such as the water/
© 1998 RAS, GJI 133, 726–740
727
crust interface or core–mantle boundary (CMB). The domain
decomposition technique developed for the construction of
exact wavefield phenomena in such physical spaces divides the
global modelling space into several subdomains separated
by interfaces at velocity discontinuities or at the boundaries
where material rheologies change (e.g. Carcione 1991). Wave
propagation is solved independently based on the correct
continuum physics describing wave motion and mechanical
behaviour in each subdomain. This method allows us to vary
the grid spacing based on the different wave speeds so that an
assigned source pulse can be appropriately sampled at a
minimum cost everywhere along the wave path. In this
paper, we extend the previous work in 2-D wave propagation
(Carcione 1991; Tessmer et al. 1992) to 3-D seismic modelling
in general anisotropic elastic media adjoining acoustic spaces.
The following section describes the principle of the multidomain, pseudo-spectral method (Chebyshev–Fourier) and its
formulation, along with the numerical treatment of boundary
and continuity conditions at the interfaces between elastic/
elastic or fluid/elastic subdomains. An optimization scheme
for the parallelization of the serial program is introduced to
render a 3-D, large-scale simulation practical in a connected
network of workstations. In Section 3, we test the numerical
accuracy of the method. Section 4 presents the simulations of
seismic wave propagation in isotropic and anisotropic models,
and begins to explore the characteristic features of waveforms
that identify the nature of the anisotropy. Finally, we construct
two-domain models in which the isotropic or anisotropic space
is in contact with a water layer to simulate wave propagation
in oceanic structures. This capability is particularly important
for studying data recorded at OBSs, because the contamination
of primary phases by large-amplitude water reverberations can
cause difficulty in analysing the OBS data (Blackman, Orcutt
& Forsyth 1995).
2 FO R M UL ATI O N O F T HE PA R A L LEL
M ULTI D O MA I N P S EU DO - S P ECT R AL
M ETH O D
The pseudo-spectral method was initially proposed to solve
the hyperbolic equations (e.g. Kreiss & Oliger 1972; Orszag
1972) and has been applied to a variety of problems in the
field of fluid dynamics since then (e.g. Canuto et al. 1988). It
was first introduced by Gazdag (1981) to model acoustic wave
propagation and by Kosloff & Baysal (1982) to model elastic
waves in 2-D isotropic media. In principle, this method employs
a set of basis functions such as Fourier series to expand field
variables and the discrete Fourier transform to calculate the
derivatives in physical equations.
Theoretically, the length spanning two nodes is the smallest
wavelength that can be resolved by the pseudo-spectral method
in a homogeneous model. In heterogeneous and/or anisotropic
media, five–10 grid points sampling the minimum wavelength
are required to construct accurate wavefields. In 2-D models,
the Fourier method with a fourth-order, time-stepping scheme
needs a factor of 3–4 less CPU time and memory storage
to perform calculations equivalent in accuracy to the fourthorder finite difference method (Daudt et al. 1989). The saving
in memory is more pronounced for 3-D problems, since the
global approximation in the pseudo-spectral method allows
coarser grids to simulate higher-frequency seismic waves in
large, inhomogeneous, even anisotropic structures. Here we
728
S.-H. Hung and D. W. Forsyth
present a multidomain, Chebyshev–Fourier method accompanying a parallel algorithm to simulate 3-D wave propagation
accurately and efficiently in both inhomogeneous and anisotropic media, particularly in oceanic structures with rapidly
varying velocity.
2.1
Basic equations
The two primary equations needed to describe elastic wave
propagation in an inhomogeneous, anisotropic medium undergoing infinitesimal deformation are the equation of momentum
conservation and the generalized constitutive law. These two
equations expressed in terms of the velocity–stress variables
are written as (e.g. Orrey 1995):
r∂ v =s + f ,
(1)
t i
ij,j
i
∂ s =c ∂ e ,
(2)
t ij
ijkl t kl
where ∂ e =(v +v )/2. v is the velocity component in the
t kl
k,l
l,k
i
ith axis, s the stress tensor equivalent to the ith component
ij
of the traction acting across the plane normal to the jth axis,
f the ith component of the body force, e the strain tensor,
i
kl
r the density and c
the fourth-order stiffness tensor with
ijkl
elastic coefficients for linear elastic solids. Using the symmetric
properties of stress and strain tensors, eqs (1) and (2) are
combined into a set of nine coupled linear first-order partial
differential equations:
∂
∂
∂
∂
[ U]=A
[ U]+B
[ U]+D
[ U]+[b]
(3)
∂t
∂x
∂x
∂x
1
2
3
where [ U]=[v , v , v , s , s , s , s , s , s ]T, [b]=
1 2 3 11 22 33 23 13 12
[ f /r, f /r, f /r, 0, 0, 0, 0, 0, 0]T and A, B, D are 9×9
1
2
3
matrices consisting of elastic constants, c , and density, r
ijkl
(see Appendix A). Subscript 3 represents the vertical dimension
in the z-direction and subscripts 1 and 2 are the horizontal
dimensions in the x- and y-directions, respectively. The temporal and spatial variations of the velocity and stress fields
can be solved from eq. (3) once the boundary and initial
conditions are specified. The boundary condition with zero
traction is imposed at the surface of the Earth. The initial
condition introduced in the body force term describes the
mechanism and time history of seismic sources; that is, the
source time function.
The advantage of using the velocity–stress formulation
instead of displacement is that the former scheme has no
derivative of displacement involved in the calculation of stress
at the free surface, giving a more accurate representation of
the traction-free boundary condition. The second-order partial
differential equation (PDE) of elastic wave motion formulated
in displacement is reduced to a first-order PDE in the velocity–
stress expression. The velocity–stress scheme is, therefore, less
sensitive to material variations since the elastic coefficients
appear undifferentiated in the wave equation (Kosloff, Reshef
& Lowenthal 1984).
(1992) introduced the application of the continuity condition
at the subdomain interfaces in a 2-D isotropic medium. Tessmer
(1995) derived the explicit forms of the characteristic variables
in general anisotropic elastic media for the boundary treatment on the free surface. Following their work we extend the
utilization of characteristic variables in computing the matching solutions of wavefields across the discontinuous interfaces
in 3-D anisotropic media, namely for the stresses and velocity
at the boundaries between solid/solid or fluid/solid domains.
We assume the ith elastic subdomain overlies the (i+1)th
elastic subdomain and the interface between these two domains
is a horizontal plane. The velocity and traction wavefields have
to be continuous when crossing a common interface. Therefore,
the outgoing characteristic variables from the ith subdomain
towards the (i+1)th subdomain need to be matched with the
incoming characteristic variables from the (i+1)th subdomain.
The nodal values of the velocity and normal traction at the
interface are respectively equalized by solving a 6×6 system
of linear equations in a matrix form:
E[ W ](n)=[ X](o) ,
(4)
where
E=
C
−b
b
41(i)
41(i+1)
−b
61(i)
b
61(i+1)
−b
81(i)
b
81(i+1)
−b
b
42(i)
42(i+1)
−b
62(i)
b
62(i+1)
−b
82(i)
b
82(i+1)
−b
b
b
43(i)
43(i+1)
−b
63(i)
b
63(i+1)
−b
83(i)
b
83(i+1)
46(i)
b
46(i+1)
b
66(i)
b
66(i+1)
b
86(i)
b
86(i+1)
[ W ](n)=[v(n) , v(n) , v(n) , s(n) , s(n) , s(n) ]T
1 2 3
33 23 13
and
b
b
47(i)
47(i+1)
b
67(i)
b
67(i+1)
b
87(i)
b
87(i+1)
b
b
48(i)
48(i+1)
b
68(i)
b
68(i+1)
b
88(i)
b
88(i+1)
D
,
]T .
, c(o)
, c(o)
[ X](o)=[c(o) , c(o) , c(o) , c(o)
−l1(i) −l2(i) −l3(i) l1(i+1) l2(i+1) l3(i+1)
The b
and b
are the mth row and nth column elements
mn(i)
mn(i+1)
of the inverse eigenvector matrix of D in eq. (3) for the ith
and (i+1)th subdomains, respectively. The superscript (n)
represents the equalized solutions of the continuous traction
and velocity at the overlapped interface between adjacent
subdomains. The superscript (o) indicates the nodal values of
the characteristic variables c with the eigenvalues −l , −l ,
1(i)
2(i)
−l in the ith subdomain and with l
,l
,l
in
3(i)
1(i+1) 2(i+1) 3(i+1)
the (i+1)th subdomain before the application of the continuity
condition. The derivation of characteristic variables is provided
in Appendix A. To avoid numerical instability when forcing
the variables to satisfy the boundary or continuity conditions,
the other three stress components not required to be continuous have to be updated on both sides of the subdomains
based on the characteristics with zero eigenvalues (Gottlieb,
Gunzberger & Turkel 1982):
[ Y ](n)=[ Y ](o)+S[Z] ,
(5)
where
2.2
Domain decomposition in 3-D anisotropic media
In the multidomain calculation, the model space is divided
into N subdomains based on wave velocity and physical
properties. Eq. (3) is solved independently in each individual
subdomain to yield a set of velocity and stress variables. These
variables at the collocation points of the domain interface need
modification to satisfy the continuity conditions. Tessmer et al.
C
D
b /b
b /b
b /b
16 14
17 14
18 14
,
b /b
b /b
S= b /b
26 25
27 25
28 25
b /b
b /b
b /b
36 39
37 39
38 39
[Z]=[s(o) −s(n) , s(o) −s(n) , s(o) −s(n) ]T
33
33 23
23 13
13
and [ Y]=[s , s , s ]T.
11 22 12
© 1998 RAS, GJI 133, 726–740
Parallel multidomain pseudo-spectral method
If the ith subdomain is the fluid and the (i+1)th subdomain
is the elastic solid, the shear stresses s and s are zero at
23
13
the common interface because the fluid cannot undergo shear
deformation and the traction needs to be continuous. The
continuity of the vertical velocity v and normal stress s
3
33
along the interface requires that the downgoing characteristic
variable with the eigenvalue −v (equivalent to the negative
p
acoustic wave speed) from the fluid subdomain is matched to
the upgoing characteristic variables with the eigenvalues l
,
1(i+1)
l
,l
from the elastic subdomain. This leads to a set
2(i+1) 3(i+1)
of 4×4 linear equations to solve for the updated wavefield
variables: the continuous vertical velocity and pressure in
the fluid or the normal stress s in the elastic domain, and
33
unnecessarily continuous horizontal velocities only for the
elastic domain,
F[R](n)=[Q](o) ,
(6)
where
C
D
b
b
b
b
41(i+1)
42(i+1)
43(i+1)
46(i+1)
b
b
b
b
62(i+1)
63(i+1)
66(i+1)
,
F= 61(i+1)
b
b
b
b
81(i+1)
82(i+1)
83(i+1)
84(i+1)
0
0
1
−1/(r v )
(i) p
[R](n)=[v(n) , v(n) , v(n) , s(n) ]T
1(i+1) 2(i+1) 3
33
and [Q](o)=[c(o)
, c(o)
, c(o)
, c(o) ]T. The s , s and
l1(i+1) l2(i+1) l3(i+1) −vp(i)
11 22
s in the elastic
subdomain are modified using the non12
propagating characteristic variables as shown in eq. (5). If the
elastic subdomain overlies the fluid layer instead, for instance
at the CMB or a fluid magma reservoir, the continuous vertical
velocity and normal stress s along the interface are obtained
33
by solving eq. (6). There the characteristic variables are
replaced either by those corresponding to the eigenvalues −l ,
1
−l , −l for the elastic subdomain or by those corresponding
2
3
to the eigenvalue v for the fluid layer.
p
2.3
The Chebyshev–Fourier method
In this study, we use the periodic Fourier basis functions to
expand the horizontal variations of wavefield variables. As
they are limited in the ability to implement explicitly the nonperiodic boundary conditions at the traction-free surface and
at the non-reflected base (e.g. Huang 1992; Lou & Rial
1995), Chebyshev polynomials are instead used in the vertical
expansion and the partial derivatives are calculated through
the expansion coefficients of the variables found by the cosine
Fourier transform and the recursion relation (e.g. Kosloff
et al. 1990). When the spatial derivative terms of eq. (3) are
computed, the wave equation is treated as a first-order, timederivative ordinary differential equation. The fourth-order
Runge–Kutta method is then employed to update the wavefields at the incremental time step. The boundary conditions
at the free surface and continuity conditions at the domain
interfaces are applied after the completion of the timeadvance step.
The use of the non-periodic Chebyshev functions for space
discretization generates decreasing grid spacings toward the
boundaries (Kosloff et al. 1990; Carcione 1996). This increases
the resolution in modelling the appearance and interaction of
different waves at the free surface or a discontinuous interface. However, the extremely small spacing next to the edges
© 1998 RAS, GJI 133, 726–740
729
introduces a stringent stability criterion for the time-stepping
interval (Kosloff & Tal-Ezer 1993). A coordinate transformation
that rescales and stretches the Chebyshev mesh is applied to
remedy this restriction and thus reduce the growth in computation time (Kosloff et al. 1990). Nonetheless, the rescaled
grid spacing near the boundary is still smaller than the uniform
spacing in the equivalent Fourier expansion.
2.4 Parallelism of the FFT-based algorithm
Numerical modelling of seismic wave propagation in 3-D
media usually suffers from a problem with memory. In the
Chebyshev–Fourier method with the velocity–stress formulation, for instance, the nine wavefield parameters and their
derivatives at two intermediate time levels in each time step,
the density and the stiffness tensor (up to 21 independent elastic
coefficients in general anisotropic media) at each grid point
need to be stored. For surface waves (Rayleigh and Love
waves), grid spacings smaller than those for body waves are
needed to obtain spatial resolution equivalent to body waves
at the same period, because surface waves propagate with
slower speeds along the free surface. The finer grids near the
surface generated from the non-uniform Chebyshev mesh may
also intensify the numerical cost by requiring smaller time
increments. Consequently, 3-D calculations running only on a
single processor not only become prohibitively slow but also
require too much memory for large-scale anisotropic models.
Parallel computation implemented in distributed memory computing machines provides an efficient alternative to tackle
problems such as this.
The framework of the FFT-based algorithms such as the
Chebyshev–Fourier method and multidomain architecture is
suited for parallel computation since most of the arithmetic
work is from FFT operations. The serial code can be parallelized
easily by breaking the numerous FFTs sequentially performed
on one processor into simultaneously running transforms
on several processors since each FFT is independent. The
major modification required for the parallel algorithm in
waveform calculation is the complete exchange (or all-to-all
communication) between processors. The message-passing
model for the 3-D FFT-based algorithm is illustrated in Fig. 1.
The 3-D model space is initially partitioned into N slab-shaped
subregions along the y-axis in Cartesian coordinates, where N
is the number of the nodes or processors. For this parallelcomputing model, each node can calculate concurrently the
first derivatives of the velocity and stress with respect to the
x- and z-directions. As for the y-derivative of wavefield variables, all processors have to complete the all-to-all communication first; that is, each node sends different pieces of
data in its local memory to all other nodes and then receives
data from all other processors (Fig. 1). The amount of data
storage remains fixed before and after the complete exchange.
Once the y-derivative is calculated, the data will be reversely
exchanged to recover the original form of data arrangement.
The program will then execute the concurrent multiplication
of coefficients and time advance.
In this study, the parallel algorithm was developed for
implementation in the IBM Scalable POWERparallel 2 system,
which connects RISC System/6000 processors via a communication network called High-Performance Switch (Stunkel
et al. 1994). The Message Passing Interface (MPI) (Gropp,
Lusk & Skjellum 1994), which provides standard libraries
730
S.-H. Hung and D. W. Forsyth
Figure 1. The message-passing model employed in parallelization
of the FFT-based pseudo-spectral scheme for synthetic waveform
calculations. The top diagram shows the configuration of data
partition in a 3-D model space with a (nx)×(ny)×(nz) grid mesh
that is distributed in N processors before (left part) and after (right
part) all-to-all communication. The bottom diagram schematically
represents the map view of the collective data movement during the
communication. The processors are narrow boxes confined by solid
lines in the drawing. A, B, … , C and D represent initially partitioned
data with (nx)×(ny/N)×(nz) data points stored in each processor
p=1, 2, … , N−1, N. They are each separated into N blocks of size
of (nx/N)×(ny/N)×(nz) data points, with each block represented
by A , B , … , C , or D (K=1, 2, … , N-1, N). During the comK K
K
K
munication, each processor exchanges its Kth data block with the Kth
processor in a diagonally symmetric fashion, so that the Kth data
blocks in all the processors will be restored into one single processor.
to assist the development of portable parallel programming
and accelerate the message passing, is used to transfer data
between the local memories of the processors. Therefore, the
communication work present in the parallel code is machineindependent and the program can be run in any multiprocessor
computer with no extra effort. The communication cost
compared to the arithmetic computation time is negligible
when the model space is large and the latency for initializing
a communication connection is not important in the long
message passing.
We ran a 64×64×65 grid model on 1, 2, 4, 8 and 16
processors to test the improvement of the parallel program in
calculation time and efficiency. The CPU time spent for 100
time-step calculations is plotted against the number of processors in Fig. 2(a). The parallel efficiency defined by [CPU
time for N processors]×N/[CPU time for single processor]
corresponding to each run is shown in Fig. 2(b). The parallel
efficiency reaches over 90 per cent except when 16 nodes are
used. Because the latency cost for communication becomes
pronounced when sending and receiving many messages with
short length, there is a significant degradation in efficiency
when 16 nodes are used. The ratio of the number of grid points
in the partitioned dimension to the number of processors
(=ny/N2, where ny is the total number of grid points in the
Figure 2. Testing the FFT-based parallel algorithm developed in this
study by running 100 time iterations over a 64×64×65 3-D model.
(a) The CPU time consumed when using 1, 2, 4, 8 and 16 processors.
( b) The parallel efficiency defined in the text for each timing-test run.
y-direction) is less than 1 in the 16 node run. To keep over 90
per cent parallel efficiency for the algorithm performance, the
number of processors should be chosen so that this ratio is at
least greater than 0.5.
2.5 Source time function and artificial boundary
condition
To illustrate signals from realistic seismic sources, we choose
the Ricker wavelet, which is based on the second derivative of
a Gaussian wavelet (Kelly et al. 1976):
f (t)=2j[1−2j(t−t )2] exp[−j(t−t )2] ,
(7)
0
0
where j=4p2/t2 and t is the pulse period at which the
p
p
amplitude spectrum is dominant, and t is the excitation time
0
of the pulse. In our study, a Ricker wavelet with frequency
centred at 1 s is used, although local sources would have higher
frequency content. Geometrically, a double-couple earthquake
or an explosion is, in general, expressed as the equivalent force
couples or the moment tensor (Aki & Richards 1980), which
© 1998 RAS, GJI 133, 726–740
Parallel multidomain pseudo-spectral method
731
in turn can be directly applied to the six stress variables in
the velocity–stress formulation. To minimize the aliasing effect
resulting from the truncated expansion of a spatial delta function
in terms of the basis functions (Özdenvar & McMechan 1996),
the point source is spatially smoothed by a Gaussian function
and spread out from the centroid location (x , y , z ) to the
c c c
neighbouring nodes in all directions, i.e.
F(x, y, z, t)= f (t) exp{−a[(x−x )2+( y−y )2+(z−z )2]} ,
c
c
c
(8)
where a is the distance of the farthest neighbouring node at
which the amplitude of the source time function is reduced by
a factor 1/e.
To remove spurious reflections and wrap-around from the
artificial boundaries of the finite numerical space, a nonreflective boundary condition based on the 1-D wave equation
approximation (e.g. Clayton & Enquist 1977) is applied on the
bottom of the model space. An absorbing boundary condition
which gradually damps the amplitude of wavefields towards
the boundaries is also implemented in a strip of the region
along the side and bottom grids to help eliminate unreal
reflections from oblique incident waves (Cerjan et al. 1985).
3 A C C U R AC Y T ES TS FO R NU M ER I C A L
S O L UTI O N S
To demonstrate the accuracy of the multidomain pseudospectral method for 3-D waveform modelling, we compare
the analytical solution of the displacement excited by a
double-couple seismic source with the synthetic seismograms
calculated by the Chebyshev–Fourier method. The homogeneous model space is discretized into a 128×128×136
mesh spanning a total distance of 50.8 km in the horizontal
directions and 54 km in the vertical dimension. The P-wave
velocity is 8 km s−1, the S-wave velocity is 4.62 km s−1 and
the density is 3.3 Mg m−3. The point source with a pure
dip-slip normal fault striking along the y-axis is located at
(x, y, z)=(24, 24, 26.8) km. The seismograms are recorded 1.2
and 15.1 km below the hypocentral depth to investigate both
near-field and far-field displacements. Fig. 3 shows the comparison between the numerical result and the exact solution of
an infinite space problem (Aki & Richards 1980). The excellent
match of the synthetic waveforms to the analytical solution at
distances less than or equal to a wavelength of the P wave as
well as several wavelengths away indicates that the pseudospectral scheme is able to model the amplitude decay of nearfield and far-field displacements and to represent precisely the
actual ground motion excited by an earthquake.
To test the accuracy of the application of the continuity
condition at the interface of a multidomain configuration, we
perform a numerical experiment in which an artificial planar
structure traverses a homogeneous space and seismic waves
propagate from the lower to the upper subdomain. The halfspace has uniform P- and S-wave velocities, the same as those
in the model of Fig. 3. The modelling space with a total extent
of 50.8 km in all three dimensions is discretized into a
128×128×129 mesh for the single-domain calculation. The
configuration of the grid mesh for the two-domain experiment
is 128×128×33 in the upper subdomain and 128×128×97
in the lower subdomain. The non-physical discontinuity that
divides the space into the two subdomains is put at a depth
© 1998 RAS, GJI 133, 726–740
Figure 3. Comparison of the numerical results of the near-field and
far-field displacements excited by a double-couple point source in an
infinite homogeneous space with the known analytic solutions. The
source with a normal fault mechanism striking parallel to the y-axis
is located at 26.8 km depth. (a) Seismograms recorded at 41.9 km
depth at distances greater than a wavelength of the P wave from the
source. ( b) Seismograms recorded at 28.0 km depth, three grid points
below the hypocentre of the seismic source at distance less than or
equal to a wavelength of the P wave. Solid lines represent the analytical
solution and crosses are the numerical solution calculated from the
Chebyshev–Fourier method. The amplitude scale in (a) is five times
larger than that in (b). At each site in this and subsequent figures,
three components of displacement are shown, with u3 the top trace,
u2 middle, u1 bottom. Insets show the geometry of the model space
and the locations of the source and receivers. The grey planes show
the horizontal planes where the stations are located. The plane
bounded by the dashed line is the plane on which the source is applied.
of 12 km. The centroid of a pure dip-slip normal fault with the
strike oriented 45° from the x to the y axis is located in
the lower domain at 14 km depth. Synthetic seismograms
recorded at the free surface and at 0.8 km below the centroid
are calculated from both single-domain and two-domain
approaches. In the two-domain case, the wavefield is calculated
separately in each subdomain and then the continuity condition
is imposed along the fake discontinuity. The synthetic seismograms from these two different calculations are almost exactly
identical (Fig. 4), indicating that the wavefield-matching routine
we apply at the overlapped grids between two domains is
correct and no artificial converted phases or reflections appear
at the non-physical interface.
The example shown in Fig. 5 illustrates that if an abrupt
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water reverberations are well simulated in the multidomain
modelling, but when only one domain with equal numbers of
grid points is used the reverberated phases are smeared out
and hardly distinguishable from the background noise. This
inaccuracy occurs because the speed of acoustic waves in the
water layer is too slow and the velocity contrast at the interface
is too large. With a single domain, the number of grid samples
per wavelength is too small to describe the propagating
wavefields within the fluid layer and the transition is too sharp
to be accurately described by the finite-order basis functions.
4 SE IS M I C WAVE P R O PA G ATI O N I N
A NI S O TR O P IC O C E AN I C S T R UC T UR ES
Figure 4. Comparison of the synthetic waveforms for a uniform
half-space using both single-domain and two-domain approaches.
(a) Seismograms recorded at the free surface. ( b) Seismograms recorded
at 14.8 km depth in the lower subdomain. Solid lines show the singledomain solution and crosses are the two-domain results. The seismic
source is a normal fault earthquake and the hypocentre is located at
14 km depth. Inset shows the geometry of the model space and the
locations of the stations and source. The dotted lines show the nonphysical planar structure at 12 km depth in the two-domain calculation.
No artificial reflections or interface waves are generated by the
introduction of a boundary with no contrasts in physical properties.
change in velocity or elastic properties exists in the model
space, the single-domain approach cannot correctly model
reflected and transmitted energy as well as wave conversions
across the discontinuous features, at least not without a
tremendous increase in the number of grid points. However,
in the multidomain method using equal or even smaller
numbers of collocation points for the same space, we can
adjust the grid mesh in different velocity zones to maintain
the minimum grid points needed for the wavelength of seismic
waves. In this example, the model space with a 63×63×106
mesh discretization spans a total distance of 31 km in each
dimension. There is a 2 km deep water layer overlying an
isotropic, elastic half-space. The P-wave velocity and density
are 1.5 km s−1 and 1.03 Mg m−3 for the water layer and
5.5 km s−1 and 2.55 Mg m−3 for the elastic half-space. Due to
the domain decomposition and the unevenly spaced Chebyshev
mesh, the grid positions are different in the vertical direction
for these two models. The centroid depth of an explosive
source is thus slightly different, and is 8.05 km for the singledomain calculation and 8.12 km for the two-domain case. The
Seismic anisotropy has been documented in many tectonic
regions such as the upwelling melting zone beneath mid-ocean
ridges, near mantle plumes, spreading oceanic lithosphere and
upper asthenosphere, mantle wedges above subduction slabs
and the D◊ region (e.g. Bowman & Ando 1987; Nishimura &
Forsyth 1989; Bjarnason et al. 1996; Kendall & Silver 1996).
Shear-wave splitting, polarization anomalies and azimuthal
anisotropy of seismic traveltimes are commonly employed to
detect the faulting and crack alignment in shallow crustal
structures (e.g. Crampin & Lovell 1991) and to study the largescale convective flow associated with plate motion and mantle
dynamics (e.g. Russo & Silver 1994).
To characterize the complication of seismic wavefields in
anisotropic structures, we perform wave-propagation simulations in our elastic models with a variety of crystallographic
symmetries that may occur in the Earth’s mantle or crust. The
first model, A, is a reference isotropic model with a P-wave
velocity of 8.94 km s−1 and an S-wave velocity of 5.16 km s−1.
The second model, B, is a transversely isotropic medium with
a horizontal symmetry axis oriented 45° from the x- and the
y-axes, i.e. in the [110] direction. The [hkl ] indicates the
outward-directed normal to the lattice plane (hkl) and the h:
is the opposite to the +h direction (Weiss & Wenk 1985).
The x-, y- and z-axes in the coordinate of the model space are
oriented parallel to the directions [100], [010] and [001],
respectively. This model can simplify the general stiffness tensor
to five independent elastic coefficients (Babuska & Cara 1991).
The third model, C, has orthorhombic symmetry with elastic
properties associated with an olivine crystal, the most abundant
mineral in the upper mantle. The a-axis, the fastest propagating
direction for the P wave, coincides with the x-axis and the
slowest b-axis is aligned along the y-axis. There are nine
independent elastic coefficients in this symmetry system. The
last model, D, has monoclinic symmetry in which two of the
three crystallographic axes are mutually perpendicular and
the third is inclined. The velocity is only symmetric to the
plane containing the orthogonal [100] and [001] axes, which
are oriented in the x- and z-directions in our calculation. The
number of independent elastic coefficients extends to 13. Most
minerals in the crust, such as feldspar and diopside, have this
symmetry.
4.1 Phase velocities of the anisotropic models
For the three anisotropic models, we solve the eigenvalues
and eigenvectors of the Christoffel matrix associated with
the stiffness tensor of an elastic medium, which yield three
phase velocities and the corresponding polarization directions
© 1998 RAS, GJI 133, 726–740
Parallel multidomain pseudo-spectral method
733
Figure 5. Comparison of the synthetic waveforms calculated by the single domain and multidomain methods when discontinuities with large
velocity contrast such as the water/solid contact are present. In this model, a 2 km water layer overlies an elastic half-space. The centroid depth of
an explosive source is 8.05 km for the single-domain and 8.12 km for the two-domain calculation. (a) A single-domain method is employed to
calculate synthetic seismograms recorded at 0 km, the free surface (shown in the top frame), 2.00 km, the water/solid interface (shown in the middle
frame) and 14.74 km depth (shown in the bottom frame). The model space is discretized to a 63×63×106 mesh for x-, y- and z-directions.
( b) Synthetic seismograms recorded at 0, 2.00 and 14.82 km depth are obtained from the two-domain calculation. The grid mesh used is 63×63×33
for the water layer and 63×63×73 for the solid space. The seismograms recorded at the same depth are normalized by the maximum amplitude
at each depth. Both cases employ equal grid points to sample the variation of the seismic pulse, but only the multidomain approach that is adapted
to modify the grid spacing based on the wave speeds can accurately model reverberated phases within the water column and converted phases at
the liquid/solid interface and free surface.
(Babuska & Cara 1991). Fig. 6 shows plots of the velocity
variation as a function of the azimuth and dip angle of the ray
path from the three propagating eigenmodes: one quasi-P wave
and two quasi-S waves. To manifest the significance of anisotropic effects on seismic observations, in Fig. 6 we also show
the wave surfaces on three orthogonal planes, XY , XZ and
Y Z, centred at the initiation point. All three models display
observable azimuthal anisotropy in P-wave arrivals. For
those points where the two wave fronts of shear waves are
furthest apart, we should obtain convincing evidence for seismic
anisotropy from shear-wave splitting.
For model B, the P and quasi-SV waves travel most slowly
in the [110] direction and fastest on the (110) plane, while the
SH wave is the slowest on this plane and the fastest along its
normal direction. Such a system may exist in sedimentary
formations resulting from a stack of horizontal, homogeneous
isotropic layers with contrasting properties, producing an
apparently vertical symmetry axis, or in the shallow crust from
the ordered alignment of fractures with a symmetry axis normal
to the faulting plane. Transversely isotropic symmetry is often
assumed in seismic studies because it reduces the number of
model parameters that have to be resolved and it is assumed
that the azimuthal dependence is averaged out either due to
locally random symmetry properties or due to averaging over
many paths covering the range of possible azimuths (e.g.
Nishimura & Forsyth 1989).
In model C, the velocities for both P and S waves are quite
different along the three crystallographic axes. The [100] axis
is the fastest, the [001] the intermediate, and the [010] the
slowest direction for the P wave. The variation of the wave
speed is dissimilar between the quasi-SV and SH phase, and
© 1998 RAS, GJI 133, 726–740
the SH proceeds much faster than the SV in most directions.
The slowest S wave propagates parallel to either the [010]
or the [001] direction, and the corresponding polarization
direction points to [001] or [010]. The fastest S wave travels
on the XZ plane, heading for the direction inclined 45° or
135° from the z-axis, and is polarized on the XZ plane normal
to the propagating direction. Such a model exemplifies the
anisotropic structure in the upper mantle, where the majority
of the constituent minerals such as orthorhombic olivine undergo
dislocation-creep deformation and preferentially orient their
a-axes to the flow direction.
The velocity dependence on the direction of propagation for
model D is much more complicated because of the oblique
crystallographic axis and fewer symmetry planes. The most
striking differences between this system and the previous ones
are that the phase velocity is only symmetric to the XZ plane,
and that the fastest direction is oblique to the slowest direction
for both P and S waves and none of them parallel to the x-,
y- or z-axis. The propagation direction for the fastest and
slowest P wave is on the symmetry XZ plane, inclined 30°
and 145° from the z-axis, respectively. The fastest and slowest
directions for the S waves are even more obscure. They are
neither mutually orthogonal nor on any planes normal to the
coordinate axes. This system may exist in subduction zones or
beneath spreading centres, where the a-axis of olivine tends to
be dipping in such a way as to accommodate the maximum
finite extension from the turning flow.
4.2 Anisotropic wavefields and synthetic waveforms
The model space for the numerical simulation is discretized
into a 128×128×101 mesh and extends 63.5 km in the
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S.-H. Hung and D. W. Forsyth
Figure 6. Azimuthal equal-area projections of anisotropic velocities ( km s−1) as a function of azimuth and dip angle for Model B with hexagonal
symmetry (or transverse isotropy) in the upper right diagram; Model C with orthorhombic symmetry in the lower left diagram; Model D with
monoclinic symmetry in the lower right diagram. In each model, velocities of quasi-P, quasi-SV and quasi-SH phases are shown in the top, middle
and bottom rows of the left column, respectively. The paired velocities in each row are projections from two opposite centres: the left is from the
top (0, 0, −1) and the right from the bottom (0, 0, 1). The right column shows the wave fronts emitted from the source on the XY (top), XZ
(middle) and Y Z (bottom) planes.
horizontal directions and 50 km in the vertical dimension. The
centroid of a strike-slip faulting earthquake with 45° orientation
(measured from the x-axis toward the y-axis) is placed at an
upper central point of the model (32, 32, 11.1) km (Fig. 7).
Fig. 8 shows snapshots of the three-component displacements
at 3.6 s after initiation of the source. We show the wavefield of
the vertical displacement at the free surface (XY plane). The
horizontal displacement in x- and y-components is respectively
shown on the vertical XZ and Y Z planes slicing through the
source point.
In an isotropic homogeneous medium such as model A, the
phase velocities of the three eigenmodes are independent of
path, and thus the wave fronts emitted from the point source
remain spherical. Shear waves are not split since the radially
and transversely polarized components propagate with the
same speed. For the transversely isotropic model B, the snapshot at the surface displays elliptical wave fronts with the long
axis aligned along the [11: 0] axis, the fast propagation direction
for the P and quasi-SV waves (that is the two outer phases),
but parallel to the [110] axis for the quasi-SH wave (that is
the innermost phase). Shear-wave splitting detects significant
time delays between the leading and late shear phases at
the azimuths parallel to the [11: 0] axis, but not in the [110]
direction where the two wave surfaces touch tangentially.
Because the phase velocities are symmetric around the x, y
and z-axes for model A and around the [110] direction for
model B, the particle motions undergone by the passing
wavefields behave similarly along all paths symmetric around
© 1998 RAS, GJI 133, 726–740
Parallel multidomain pseudo-spectral method
Figure 7. The geometry of the model space, seismic source and
recording stations for the simulation of wave propagation in the
velocity models shown in Fig. 6. Solid triangles indicate the four
receivers located at the free surface, denoted by S1, S2, S3 and S4. The
ray paths are shown by solid lines. The three grey planes are the
places where the snapshots in Fig. 8 are recorded. The focal sphere
shows the strike-slip fault at the hypocentre projected onto the XY
plane. Black quadrants indicate compressional P-wave motions.
these axes, and they are recorded on the surface with identical
wave shapes and amplitudes. Therefore, for these two models
the snapshots of the wavefields on the XZ and Y Z planes are
equivalent in shape but opposite in polarity because of the
orientation of the double-couple source.
In orthorhombic model C, the P and S waves travelling
along the x-axis are faster than along the slowest y-axis and
the intermediate z-axis, resulting in three distinct, non-circular
wave fronts on these planes. In addition to the diversity of
wave-front shapes linked to azimuthal anisotropy in phase
arrival times, the change of wavefield amplitudes is also
different from the previous models. On the XY and XZ planes,
two horizontal bitangent lines along the x-direction, which are
individually tangent to the wave-front curves at two points,
exist for the quasi-SH phase (see Fig. 6). These correspond to
the SV waves propagating with a local maximum speed along
the nodal planes of the radiation energy (45° from the x-axes),
where both the P and S waves have minimum amplitudes in
the vertical component. At the free surface, the P-wave amplitudes are as small as those in models A and B, but the excited
S-wave amplitudes are quite large in these directions. The most
interesting features in the S wavefields are triangular-shaped
regions of upward (negative) displacement and cusps pointing
in the +x and −x directions, associated with the interference
of the quasi SV and SH waves. The wave surfaces from these
two shear phases graze each other at these places. The amplitudes of the S waves in the x-component are relatively small
on the XZ plane except near the bitangent points of the faster
SH wave. The velocity variations in the y- and z-directions are
more subdued, leading to more circular-like wave fronts on
the Y Z plane. Because the velocity contrast between the SV
and SH phases is small on this plane, the split shear waves are
not separated enough to be detected in the wavefield snapshot.
For monoclinic model D, the snapshots of displacement are
complex due to the low degree of symmetry and the intricate
shear-wave splitting. The wave surfaces on the XY plane
through the source point are symmetric to the x- and y-axes
(Fig. 6), but only the symmetry to the x-axis is preserved at
the free surface. The upgoing P waves propagate faster towards
the +x- than the −x-direction, but the S waves are slower
© 1998 RAS, GJI 133, 726–740
735
towards the +x-direction, which yields the non-symmetric
distortion of the wave fronts. The geometries and amplitudes
of the wavefields are entirely different in the two x-directions.
Because the crystallographic axis [010] is inclined from the
y-axis, the wave surfaces on the XZ plane are distorted in a
tilted configuration. The downgoing P waves are faster towards
the −x- than the +x-direction. The S waves behave in the
opposite way. For the downgoing S waves, the SH is the
leading shear phase in the +x quadrant. The wave surfaces
from the SV and SH phases intersect in the −x quadrant and
thus the SV phase moves ahead of the SH (Fig. 6). Except for
polarity reversal, the wave fronts on the Y Z plane are symmetric with respect to the XZ plane, the only symmetry plane
for this system. When the S waves propagate in the Y Z plane
along the y-direction, the SV and SH phases are split more
than in any other direction because the SH propagates with
the local maximum speed while the SV propagates with the
minimum. The disturbed wavefields are kinked near the places
where the two shear phases arrive close together and interfere
with each other.
In Fig. 9 we show azimuth-dependent variabilities of arrivals
and waveforms in synthetic seismograms for these four
models. Four receivers are placed on the free surface at the
same distance of 15.1 km from the centroid of the source. The
locations of the stations and their ray paths are shown in
Fig. 7. No azimuthal anisotropy is shown in the transversely
isotropic model because the azimuths of these four particular
stations are off from the symmetry axis by the same degree
and the seismic waves go through the same velocity variation
along the ray paths. Although there is not much difference in
P waveforms between the isotropic and transversely isotropic
models at these azimuths, the S waveforms show diagnostic
distinctions. The single, one-cycled pulse shown in the isotropic
model is split into two on the vertical and horizontal components of the transversely isotropic model. In the orthorhombic model, the P- and S-wave velocities are symmetric
around the x- and y-axes, so there are no differences in
waveforms or arrival times between Stations 1 and 2 or
between Stations 3 and 4 (except for changes in polarity on
the horizontal components). However, the azimuth of the ray
path for Station 2 is near the fastest axis while Station 3 is
close to the slowest axis, so there are significant differences
between P and S arrival times and S waveforms for this station
pair. Shear-wave splitting is clearly seen on the two horizontal
components of Stations 1 and 2. The monoclinic model shows
variations of P-wave arrival time and non-symmetric, complicated waveforms at all four stations because the Y Z plane
(100) is not the symmetry plane. The P wave travels fastest
and the S wave propagates most slowly towards Station 1,
resulting in the largest S–P differential times. The effect at
Station 2 is just the opposite because the P wave travels most
slowly while the S wave is much faster.
4.3 Wave propagation in the space containing a water
layer
To illustrate that the multidomain method can effectively
simulate energy partitions in models with discontinuities,
we add a 2 km deep water layer on the top of an isotropic
(Model A) and an anisotropic half-space with orthorhombic
symmetry (Model C). A normal-faulting earthquake with 45°
strike is excited at location (32, 32, 11) km. Fig. 10 schematically
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S.-H. Hung and D. W. Forsyth
Figure 9. Comparison of waveforms of the four models recorded on the free surface at the same distance, but at several different azimuths. Dashed
lines represent the vertical components of displacement. Horizontal components pointing to the x- and y-directions are indicated by dotted and
solid lines, respectively.
Figure 10. The geometry of the model space, seismic source and recording stations for the simulations of wave propagation in the elastic models
(A and C) overlain by a water layer. Solid triangles indicate the 10 receivers located at the water-solid interface, denoted by S1, S2, … , S9 and S10.
The ray paths are shown by solid lines. The dark grey region represents the 2 km water layer. Three light grey planes are the places where the
snapshots in Fig. 11 are recorded. The focal sphere shows the normal fault exerted at the hypocentre projected onto the XY plane. Black quadrants
indicate compressional P-wave motions. The side view along the strike shows the slip motion on the fault plane.
shows the geometry of the model space and the focal mechanism of the double-couple source. The snapshots of the
vertical displacement at the water/solid interface and the XZ
and Y Z planes slicing through the centroid of the source are
shown at 5 s after the origin time in Fig. 11. On these three
planes, the circular S wave front for the isotropic model and
the elliptical wave surface for the orthorhombic model are
distinguishable. The leading wave front appearing on the XY
plane that advances to the S wave, occurring only at longer
distances, is the inhomogeneous P wave. The wave front that
forms in the centre region of the XY plane is the reverberated
phase, spwP, which is converted from S at the water/solid
interface and bounces once in the water layer. The multiple
water reverberations are clearly shown in the snapshots of the
XZ and Y Z planes.
In Fig. 12, we show the vertical displacement and pressure
recorded at the water/solid contact with a variety of azimuths and distances. The positions of the receivers and the
© 1998 RAS, GJI 133, 726–740
Parallel multidomain pseudo-spectral method
737
Figure 12. Comparison of synthetic seismograms for the isotropic and the orthorhombic half-spaces overlain by a 2 km water layer. Pressure and
vertical displacement are recorded at the medium interface as a function of distance and azimuth. The former is shown at the top of each paired
trace by a solid line and the latter at the bottom by a dashed line (a) for the isotropic model; (b) for the orthorhombic model. The number in
degrees shown on each paired record indicates the azimuth of the ray path.
corresponding ray paths are shown in Fig. 10. Differences
in amplitude and waveform between these two models are
clearly illustrated on both displacement and pressure records,
particularly for the shear wave and S-to-P reverberated phases.
For instance, the amplitude of the S wave at station S2 for the
orthorhombic model is small and nearly invisible compared
to that in the isotropic model. Shear waves are split when
seismic waves do not propagate along the symmetry axes, such
as those arriving at the stations S4, S5 and S6. The head-wave
P phases that appear between direct P and S at horizontal
distances greater than 17 km are converted from the SV
component when the incident angle exceeds 60° and is
beyond the critical angle for the reflected P wave. In addition,
the water-reverberated phases such as pwP and spwP are well
constructed. The multiple reverberations bouncing within the
water column are separated by an equal time delay of about
2.6 s for the two-way traveltime in the 2 km water layer. The
polarity of the pwP phase is reversed on the free surface relative
to the direct P phase in pressure, but not in displacement.
Therefore, the multiple pwP and swP phases change polarity
in the pressure recording when bouncing once in the water
layer. The much larger amplitude of pwP relative to P in
pressure is due to the constructive superposition of incident
and reflected energy at the water/solid interface. For displacement, however, the incident P wave from the water layer
reflected at the interface reverses in polarity. Destructive
interference between the incident and reflected waves at the
© 1998 RAS, GJI 133, 726–740
seafloor causes a low amplitude of the multiple reverberations
on the vertical component. These observations are consistent
with predictions of simple ray theory and reflectivity synthetics
(Blackman, Orcutt & Forsyth 1995).
5
CO NCL US I ON S
The simulation results demonstrate that the multidomain
pseudo-spectral method is able to calculate accurately synthetic seismograms and propagating wavefields in anisotropic
structures with a free surface or with large velocity gradients
such as that associated with the water/crust boundary. The
multidomain approach, which uses an appropriate domain
decomposition scheme and implements the continuity condition along discontinuous contacts provides the flexibility to
change grid spacings based on the variation of wave speed
and the requirement of the minimum number of grid points
needed to sample a complete wavelength. Parallelization of
the serial code to run in high-performance parallel computers
makes the computation of 3-D simulations or large-scale
problems practical, so that the discrete numerical solutions
become increasingly powerful and competitive for the seismological applications of full-waveform modelling. The parallel
algorithm using the message-passing model supported by the
standard Message-Passing Interface library is not restricted
to run only on the SP2 computer. It is portable and can be
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S.-H. Hung and D. W. Forsyth
implemented on many other processor-connected machines for
future applications.
The full-waveform modelling calculated from the Chebyshev–
Fourier method not only generates accurate synthetic seismograms recorded at the free surface, but also provides snapshot
displays to investigate effectively the complexity of wave-front
geometry and phase conversions in inhomogeneous and anisotropic media. The global discrete approximation of variables
in the pseudo-spectral method does not generate the numerical
dispersion and fictitious anisotropy characteristic of finite
difference methods, which is particularly important for the
study of anisotropic wave propagation. The modelling results
demonstrate that the wavefields observed at various azimuths
and distances provide critical constraints on differentiating
seismic anisotropy and heterogeneity. The synthetic seismograms for the anisotropic models with different crystallographic symmetries show distinguishable signals in waveform
amplitude and traveltimes that are full of promise for resolving
complicated elastic, anisotropic structures. These simulations
take a first step towards performing more realistic modelling
of seismic wave propagation. The synthetic results will assist
in interpreting observed data and in extracting the maximum
amount of information to search for the robust anisotropic
structure and velocity variations in many tectonically active
regions.
A CK NO W L ED GM EN TS
We wish to thank Jeff Orrey for providing his thesis and
a 2-D isotropic code as a starting place to develop our
3-D, multidomain pseudo-spectral scheme for generalized
anisotropic and heterogeneous media. We also thank Marc
Parmentier and Karen Fischer for their comments at the
beginning of this work. The thorough and critical reviews from
Colin Thomson and two anonymous referees helped us to
improve the manuscript significantly. S-HH benefitted from
discussion with Paul Fischer on the parallelization of the 3-D
serial code. Examples shown in the paper were run on the
24-node IBM SP2 machine sponsored by the Center for Fluid
Mechanics (CFM) of the Department of Applied Mathematics
at Brown University. Constantinos Evangelinos and Sam
Fulcomer from CFM gave kind assistance in managing parallel
jobs on the SP2 computer. This research was supported by
the National Science Foundation under grants OCE-9314489
and OCE-9402375.
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A P P EN DI X A: C H A R A C TE R IS TI C
VA R IA B LES FO R M ULT ID O MA I N S PA C E
The matrices A, B, D containing the elastic coefficients and
density of the medium have the following explicit forms:
C D
1/r 0 0 0
0
A=
C
11
C
16
C
15
C
21
C
26
C
25
C
31
C
36
C
35
C
41
C
46
C
45
C
51
C
56
C
55
C
61
C
66
C
65
0
0
0
1/r
0
0 0 0
0
0 0 0 1/r
© 1998 RAS, GJI 133, 726–740
0
C D
C D
0
0
0
0
0 1/r
0 1/r 0
0
0
0
0 1/r 0
0
0
C
B=
16
C
12
C
C
14
26
C
22
C
24
C
36
C
32
C
34
C
46
C
42
C
44
56
C
52
C
54
66
C
62
C
64
C
C
0
0
0
739
,
0
0 0
0
0
0 0
0
1/r
0
0
0
0
0
0 0 1/r
1/r 0
C
C
C
15
14
13
D=
C
C
C
25
24
23
C
C
C
35
34
33
,
(A1)
0
C
C
C
45
44
43
C
C
C
55
54
53
C
C
C
65
64
63
where C =C is a common 6×6 matrix expression for the
mn
nm
fourth-order stiffness tensor c using symmetric properties of
ijkl
c =c =c =c . The formal relation between C and
ijkl
jikl
ijlk
klij
mn
c is C =c with m=i=j if i=j and m=9−i−j if i≠j,
ijkl
mn
ijkl
and n=k=l if k=l and n=9−k−l if k≠l (Babuska &
Cara 1991).
The characteristic variables for the wave equation are
obtained from 1-D analysis in which only derivatives to the
normal direction of the surface are considered (e.g. Tessmer
et al. 1992). Therefore, eq. (3) becomes
∂[ U]
∂[ U]
=D
.
∂t
∂x
3
(A2)
Diagonalizing the matrix D and multiplying its inverse
eigenvector matrix T−1 on both sides of eq. (A2), we get
∂[ V ]
∂[ V ]
=L
,
∂t
∂x
3
0
,
(A3)
where [ V]=T−1[ U] is the vector of characteristic variables,
L=T−1CT is the diagonal matrix containing the eigenvalues of the matrix D, and T is the matrix composed of
the column eigenvectors of the eigenvalues (0, 0, 0, l , −l , l ,
1
1 2
−l , l , −l ).
2 3
3
For the fluid domain, the acoustic wave equation is solved
instead to obtain the velocity and pressure. The field variables
[ U] that appears in eq. (3) becomes [v , v , v , −P]T. The
1 2 3
740
S.-H. Hung and D. W. Forsyth
matrices A, B and D are simplified to
A=
C D C D
C D
0
0 0 1/r
0
0 0
0
D=
0
,
0
0
0
0
0
0 1/r
0
0
B=
0 0
0
0
0
0
K 0 0
0
0 K 0
0
0 0
0
0
0 0
0
0
0 0
0
1/r
0 0 K
,
0
Following the similar 1-D characteristic variable analysis
for elastic media, the matrix D in the fluid domain has two
zero eigenvalues and two non-zero eigenvalues that are equal
to ±v (=√K/r), where v is the speed of the acoustic waves.
p
p
The characteristic variables are
,
(A4)
A B
v −P/(rv )
3
p
v +P/(rv )
p .
[ V ]= 3
v
1
v
2
(A5)
where K is the bulk modulus.
© 1998 RAS, GJI 133, 726–740